Well, although we got that big hole in the cupboard sealed up, I still have mice. We've searched through the house, and have found two more large holes - one around the pipes in the furnace room and one around the pipes behind the washer and dryer. These are not the small little holes you might find around normal plumbing stacks - say the size of a quarter - these are closer to being a foot square. My brother-in-law and I have a date this morning to see if we can seal up all the holes.
He was asking me whether I'd consider having a cat. Not unless it was the only possibility. I'm not a cat person, and I really don't want a pet. So we'll work on sealing up the holes first.